According to the governor of southern Somalia's Lower Shebelle region, Abdukadir Nur Sidi, the attack took place late on Monday night. The governor could not confirm Godane's death during the attack.
"The drone launched several attacks where leaders of Al Shabaab and their fighters were holding a secret meeting," Sidi told Somalian broadcaster RBC.
The attack took place in a village between Lower Shebelle and Dhey Tubaako, around 190 km south of the Somali capital, Mogadishu.
"We received information that leaders of Al Shabaab, including Ahmed Godane, were meeting in this location," the governor added.
It is not yet clear whether the attack killed Godane, also known as Abu Zubair.
It is not yet clear whether the attack killed Godane
There were no other details immediately available on the operation which came days after African Union troops and the Somali Army took over several strategic villages previously controlled by the extremist movement in the same region.
The Al Shabaab movement, which announced in 2012 its loyalty to the Al Qaeda, aims to impose a radical Islamic state in Somalia. It was designated as a terrorist organisation by the US government in 2008.
Its militias control vast areas of southern and western Somalia, while its image on the international scene became more tarnished when it claimed responsibility for attacking a shopping mall in in the Kenyan capital Nairobi, that left 67 people dead.
The extremist organisation has been fighting against the transitional government and the African Union forces that support it since 2006.
Somalia has suffered chaos and anarchy following the 1991 ouster of dictator Mohamed Siad Barre. Since then, the country has remained largely in the hands of extremist Islamist militias, warlords and armed criminal gangs.